Recent Studies on Lp-Norm Estimation

Abstract: When estimating the parameters in a linear regression model, the method of least squares (L^-norm estimator) is often used. When thè residuals are independent and identically normally distributed, the least squares estimator is BLUE as well as equivalent to the maximum likelihood estimator. However, the least squares estimator is known to be sensitive to departures from the assumption of normally distributed residuals. In a variety of applications there are theoretical as well as empirical evidences that the residuals display distributional properties different from those of normal distributions. It is therefore desirable to develop alternative estimators. Members of the class of Lp-norm estimators have here been proposed as alternatives to the least squares estimator.In this monograph, questions concerning the existence, uniqueness and asymptotic distributions of L^-norm estimators are discussed. It is seen that an L^-norm estimate will always exist and it will be u-nique for 1 < p < ». For p = 1 a necessary and sufficient condition on the regressors for unique estimation is given. The question of u-niqueness in large samples is also discussed. The asymptotic distribution of Lp-norm estimators i shown to be normal, for sufficiently small p. When selecting an L^-norm estimator, a procedure based on the asymptotic variance is proposed.Finally, the possibilities to construct L -norm based estimatorsPfor estimating multicollinear regression models and models with serially dependent residuals are discussed. L -norm based methods forPestimating interdependent systems are also considered.